18-5 The First Law of Thermodynamics#
Prompts
State the first law of thermodynamics. What do \(Q\) and \(W\) represent? What are their sign conventions?
A gas expands at constant pressure. Is the work \(W\) done by the gas positive or negative? How would you find \(W\) on a \(p\)–\(V\) diagram?
In an adiabatic process, \(Q = 0\). How does the first law simplify? If the gas expands adiabatically, what happens to its internal energy?
A system goes through a cyclic process and returns to its initial state. What is \(\Delta E_{\text{int}}\)? How does \(Q\) relate to \(W\)?
Why is \(\Delta E_{\text{int}}\) path independent while \(Q\) and \(W\) are path dependent?
Lecture Notes#
Overview#
The first law is energy conservation for thermodynamic systems: the change in internal energy equals heat added minus work done by the system.
\(\Delta E_{\text{int}} = Q - W\)—heat in and work out both affect internal energy.
Internal energy \(E_{\text{int}}\) is a state function (depends only on the state); \(Q\) and \(W\) are path dependent.
The first law#
\(E_{\text{int}}\): internal energy—kinetic and potential energy of atoms/molecules. A state function; \(\Delta E_{\text{int}}\) depends only on initial and final states.
\(Q\): heat transferred to the system. \(Q > 0\) = absorbed; \(Q < 0\) = released.
\(W\): work done by the system. \(W > 0\) = gas expands; \(W < 0\) = gas is compressed.
Physical meaning: Internal energy increases when heat is added or when work is done on the system (compression). It decreases when heat leaves or when the system does work (expansion).
Work done by a gas#
For a gas that changes volume from \(V_i\) to \(V_f\):
On a \(p\)–\(V\) diagram, \(W\) is the area under the curve from initial to final state.
Expansion (\(V_f > V_i\)): \(W > 0\)—gas does work on surroundings.
Compression (\(V_f < V_i\)): \(W < 0\)—surroundings do work on gas.
Special processes#
Process |
Condition |
First law reduces to |
|---|---|---|
Adiabatic |
\(Q = 0\) |
\(\Delta E_{\text{int}} = -W\) |
Constant volume |
\(W = 0\) |
\(\Delta E_{\text{int}} = Q\) |
Cyclical |
\(\Delta E_{\text{int}} = 0\) |
\(Q = W\) |
Free expansion |
\(Q = 0\), \(W = 0\) |
\(\Delta E_{\text{int}} = 0\) |
Adiabatic: no heat flow; expansion → internal energy decreases (gas cools).
Constant volume: no work; all heat goes into changing \(E_{\text{int}}\).
Cyclical: system returns to initial state; net heat in = net work out.
Free expansion: gas expands into vacuum in insulated container; no heat, no work → \(E_{\text{int}}\) unchanged.
Poll: Isothermal process—heat and work
In an isothermal process, the gas does 50 J of work on its surroundings. How much heat was added to the gas?
(A) A positive amount < 50 J
(B) A positive amount = 50 J
(C) A positive amount > 50 J
(D) Zero joules
(E) A negative amount
Poll: Isobaric process—heat vs work
In an isobaric process, the gas does 50 J of work on its surroundings. How much heat was added to the gas?
(A) A positive amount < 50 J
(B) A positive amount = 50 J
(C) A positive amount > 50 J
(D) Zero joules
(E) A negative amount
Poll: Isochoric process—heat and work
A system undergoes an isochoric process in which its internal energy increases by 20 J. Which is correct?
(A) Heat: none; Work: 20 J done on system
(B) Heat: none; Work: 20 J done by system
(C) Heat: 20 J removed; Work: none
(D) Heat: 20 J added; Work: none
(E) Heat: 40 J added; Work: 20 J done by system
Path dependence
\(Q\) and \(W\) depend on how the system goes from initial to final state. \(\Delta E_{\text{int}}\) does not—it is the same for any path between the same two states.
Poll: Heat in, temperature down—volume
A gas undergoes a process in which 30 J of heat is added to the gas yet its temperature goes down. What can we conclude about the volume?
(A) The gas definitely expands
(B) The gas definitely contracts
(C) The gas could have expanded or contracted; not enough info
Poll: Adiabatic expansion
A gas expands adiabatically (\(Q = 0\)). Its internal energy:
(A) Increases
(B) Decreases
(C) Stays the same
Example: First-law bar chart to pV process#
Example: First-law bar chart to pV process
Take the following first-law energy bar chart and sketch a possible process on a \(p\)–\(V\) diagram.
[FIGURE: Energy bar chart showing relative magnitudes of \(\Delta E_{\text{int}}\), \(Q\), and \(W\)—e.g., \(Q > 0\), \(W > 0\), \(\Delta E_{\text{int}} = Q - W\) with appropriate relative sizes]
Solution: Use \(\Delta E_{\text{int}} = Q - W\) to check consistency. Identify whether expansion or compression (sign of \(W\)), heating or cooling (sign of \(\Delta E_{\text{int}}\)). Sketch path(s) that give the right area under the curve for \(W\).
Example: Work along different paths#
Example: Work along different paths
The figure shows two processes by which 1.0 g of nitrogen gas moves from state 1 to state 2. The temperature of state 1 is 27°C.
[FIGURE: pV diagram with state 1 and 2; path 1→3→2 (e.g., isobaric then isochoric) and path 1→4→2 (e.g., isochoric then isobaric); axes labeled]
What is the net work done by the gas (a) along path 1→3→2? (b) along path 1→4→2?
Solution: \(W\) = area under each path. Rectangular paths: sum of rectangle areas. Note \(W\) differs for the two paths.
Summary#
\(\Delta E_{\text{int}} = Q - W\)—first law; \(W\) = work done by the system.
\(W = \int p\,dV\)—area under the curve on a \(p\)–\(V\) diagram.
Adiabatic (\(Q = 0\)): \(\Delta E_{\text{int}} = -W\); expansion → \(E_{\text{int}}\) decreases.
Cyclical: \(\Delta E_{\text{int}} = 0\) → \(Q = W\).